14 LOCALLY CONVEX SPACES 71
that, if A is convex and (xt)i$t$a is any finite sequence of points of A and
n n
is any sequence of n real numbers i>0 such that V A, = 1, we have Y A, x( e A.
1*1 *=i
(a) Let E, F be vector spaces over R and let/be an affine linear mapping of E into F.
Then the image under / of any convex subset of E is convex, and the inverse image
under/of any convex subset of F is convex. In particular, if g ^ 0 is a linear form on
E, then the half-spaces defined by one or other of the relations g(x) > a, g(x) ^ a
(where a is any real number) are convex sets.
(b) In a vector space E over R, the intersection of any family of convex sets is convex.
If A, B are convex subsets of E and a, jS are any real numbers, then the set aA + /?B
is convex.
(c) Let (Ea)ae i be a family of vector spaces over R and let E = fj Ea be their product.
a
For each a, let Aa be a nonempty subset of E«. Show that A — f][ Aa is convex if and
a
only if each Aa is convex.
(d) In a topological vector space E, the closure A and the interior A of a convex set
A are convex. If, moreover, A is not empty, then A is the closure of A, and A the
interior of A.
(e) In a topological vector space E, for a half-space {x : g(x) i> a} (where g is a non-
zero linear form on E) to be either closed or to have an interior point, it is necessary
and sufficient that g should be continuous; in which case the hyperplane g(x) = a
is the frontier of the half-space.
12. (a) Let E be a vector space over R and p a semi norm on E. Show that the set of
x e E such that p(x) < 1 (resp. p(x) ^ 1) is an absorbing symmetric convex set. Con-
versely, let A be an absorbing symmetric convex set in E, and for each JteE let
p(x) — inf p. Show that p is a seminorm on E and that A contains the set of
p>0
jcepA
points such that p(x) < 1 and is contained in the set of points such that p(x) rg 1.
(b) Let E be a topological vector space over R, and p a seminorm on E. Then the
set A = {x e E : p(x) <£ 1} is closed if and only if p is lower semicontinuous on E.
Also A has an interior point if and only if p is continuous at the point 0, and p is then
continuous throughout E.
(c) Deduce from (b) that a topological vector space E is locally convex if and only
if there exists a fundamental system of convex neighborhoods of 0 in E. The topology
of a metrizable locally convex space can be defined by an at most denumerable family
of seminorms.
13. Let E be a real vector space. If A is any subset of E, the convex hull of A is defined to
be the smallest convex set containing A. If E is a topological vector space, the closed
convex hull of A is defined to be the smallest closed convex set containing A; it is also
the closure of the convex hull of A.
(a) Show that the convex hull of A is the set of all points 2 ^«x* > where (x«) is any
X
finite family of points of A, and the Aa are real numbers ^0 such that £ Aa » 1.
a
(b) Suppose that E is a metrizable locally convex space. Show that the convex hull
in E of a precojnpact set is precompact. If E is a Frechet space, the closed convex hull
of a compact set is compact (cf. Section 13.4, Problem 13).
14. Let E be the real vector space of regulated functions / on 1 = [0,1], continuous
on the right at all / < 1 and such that/(I) = 0. For each integer n > 0, let V« be thee extended to a non-